On new common fixed point theorems via bipolar fuzzy b-metric space with their applications

This research work is devoted to investigating new common fixed point theorems on bipolar fuzzy b-metric space. Our main findings generalize some of the existence outcomes in the literature. Furthermore, we illustrate our findings by providing some applications for fractional differential and integral equations.


Introduction and basic materials
Sklar and Schweizer first developed a continuous triangular norm [1] in 1960.Following that, Zadeh [2] presented the fuzzy set theory in 1965.Michalek and Kramosil [3] introduced the fuzzy metric space (FMS) in 1975 using the idea of fuzziness and the continuous t-norm.The fuzzy concept to distance is based on the assumption that the distance between any two points, which we must estimate or find, need not always be defined by a real number; rather, it is a fuzzy concept.Veeramani and George [4] updated the FMSs definition in 1994.Karamosil and Michalek [3], Grabeic [5] extended the Banach fixed point theorem to FMSs.The fuzzy Banach contraction theorem was extended to FMS by Gregori and Sapena [6] in the sense of George and Veeramani's [4].Roy, Saha, George, Gurand and Mitrović [7] introduced bipolar p-metric spaces and proved fixed point theorems.Fixed point results have been established in the setting of bipolar metric space by Karapınar and M. Cvetković [8].It examined into how these results related to analogous fixed point results in metric space, ultimately obtaining equivalency.Concepts of Pompeiu-Hausdorff bipolar metric, multivalued covariant, and contravariant contraction mappings in bipolar metric spaces were introduced by Mutlu, O ¨zkan, and Gu ¨rdal [9].Bipolar metric spaces, as generalized by Mutlu and Gurdal [10], provide a novel framework for measuring the farness between component of distinct two sets.Recently, fixed point theorems were proven, and fuzzy bipolar metric space was presented by Bartwal et al. [11].Bipolar controlled fuzzy metric spaces were first proposed by Tiwari and Rajput [12], which also demonstrated fixed point theorems.Further articles which relate can be seen [13][14][15][16][17][18].Ramalingam et al. [19], presented the fuzzy bipolar b-metric space in 2023 and used the triangular property to prove some fixed point theorems without continuity.

Main results
This section begins with the common fixed point theorem on BFBMS under covariant mappings.
Therefore, the bisequence Therefore, axiom (ii) of Theorem 2.1 also fulfills by T and S. According to Theorem 2.1, one finds that T and S have a UCFP, i.e., φ = 1.
Next, under contravariant mappings, the common fixed point theorem on BFBMS is presented.

Application to integral equations
In this section, we use the terms and conditions of the Theorem 2.1 by studying a solution of integral equations.Theorem 3.1.Consider the following coupled integral equations where T2) There exists a continuous function y : L 2 1 [ L 2 2 !½0; 1Þ, and σ 2 (0, 1) s.t.

Application to fractional differential equations
Physical systems with continuous distributions or interactions can be modeled and investigated with the help of fractional differential equations (FDEs).They are often used in engineering research to extract relationships between numbers or to provide a more detailed description of phenomena than differential equations alone can.They provide a framework to grasp various engineering systems' intricate interactions and behaviors.Implicit fractional differential equations (IFDEs) have various potential uses in engineering research.We demonstrate that there is a single solution to the IFDE in this section.In engineering, differential equations of this type are commonly used.They are necessary for material science research, heat exchange, field magnetic assessment for radars, structural evaluation, mechanisms for control, digital circuits assessment, mechanical design fatigue and circulation of fluids simulation, and data processing operations.They are also useful in geophysics, non-destructive testing, medical imaging, and inverse issues related to ophthalmology and acoustically for propagation of waves and diffraction studies.These equations provide a flexible framework for understanding and assessing continuous interactions and distributions in various engineering fields [21][22][23][24].Younis and Abdou [25] innovative method by combining concepts from graph mappings, Kannan mappings, and fuzzy contractions to produce a completely new idea known as Kannan-graph-fuzzy contraction and applications to engineering science.For more details, we refer readers to these works [26,27].In what follows, we prove the uniqueness of solution for the following fractional differential equations in the sense of Caputo derivative.
For more details see this work [28].